Tensor products of Minimal Holomorphic Representations (Q2761180)

Let \(G/K\) be an irreducible Hermitian symmetric space with genus \(p\). It can be realized as a bounded convex domain \(D\) in a complex \(n\)-dimensional vector space \(V\) with \(G\) realized as the identity component of the group of biholomorphic mappings of \(D\) and \(K\) the isotropy subgroup of \(0\in D\). The author first recalls some notions about bounded symmetric domains such as the weighted Bergman space \(H^\nu\) with \(\nu>p-1\) and the Wallach set \(W(D)\) of \(D\). The group \(G\) acts unitarily on \(H^\nu\) and it gives an irreducible unitary representation of \(G\). The author gives the irreducible decomposition of the tensor product \(H^\nu\otimes \overline {H^\nu}\) for the first nontrivial discrete point \(\nu\) of \(W(D)\). The main idea is to consider the restriction operator \(R\) from \(H^\nu \otimes\overline {H^\nu}\) to the space \(C^\infty (D)\). The operator \(R\) intertwines the tensor product action with the regular action of \(G\). The irreducible decomposition of the tensor product \(H^\nu \otimes \overline{H^\nu}\) is derived by the study of the Shimura system of invariant differential operators on \(C^\infty(D)\) and their annihilation property. As a consequence of the results, the author asserts that we discover some new spherical unitary representations of \(G\).